Characterization of Hopf Quasigroups

In this paper, we first discuss some properties of the Galois linear maps. We provide some equivalent conditions for Hopf algebras and Hopf (co)quasigroups as its applications. Then let $H$ be a Hopf quasigroup with bijective antipode and $G$ be the set of all Hopf quasigroup automorphisms of $H$. We introduce a new category $\mathscr{C}_{H}(\alpha,\beta)$ with $\alpha,\beta\in G$ over $H$ and construct a new braided $\pi$-category $\mathscr{C}(H)$ with all the categories $\mathscr{C}_{H}(\alpha,\beta)$ as components.